The Flowering of Applied Mathematics in America

Mathematicians are notoriously bad historians; they describe the development of an idea as it should logically have unfolded rather than as it actually did, by fits and starts, often false starts, and buffeted by forces outside of mathematics. In this sketchy account I will describe the twists and turns as well as the thrusts of applied mathematics in America. Applied mathematics is alive and well in America today. Looking at the 18 lectures [presented at the Centennial Meeting of the American Mathematical Society] chosen to describe the frontiers of research, we see that topics include physiological modeling, fluid flow and combustion, computer science, and the formation of atoms within the framework of statistical mechanics. We also see that the subject of one lecture and the starting point of several others are physical theories; the conclusions reached are of interest to physicists and mathematicians alike. This was not always so. For a few decades, the late 1930s through the early 1950s, the predominant view in American mathematical circles was the same as Bourbaki's: mathematics is an autonomous abstract subject, with no need of any input from the real world, with its own criteria of depth and beauty, and with an internal compass for guiding further growth. Applications come later by accident; mathematical ideas filter down to the sciences and engineering. Most of the creators of modern mathematics-certainly Gauss, Riemann, Poincare, Hilbert, Hadamard, Birkhoff, Weyl, Wiener, and von Neumann-would have regarded this view as utterly wrongheaded. Today we can safely say that the tide of purity has turned; most mathematicians are keenly aware that mathematics does not trickle down to the applications, but that mathematics and the sciences, mainly but by no means only physics, are equal partners, feeding ideas, concepts, problems, and solutions to each other. Whereas in the not so distant past a mathematician asserting "applied mathematics is bad mathematics" or "the best applied mathematics is pure mathematics" could count on a measure of assent and applause, today a person making such statements would be regarded as ignorant. How did this change come about? Several plausible reasons can be discerned. But first a bit of selective history. World War II, a watershed for our social institutions, concepts and thinking, permanently changed the status of applied mathematics in America. That is not to say that there was no worthwhile applied mathematics in America before 1945; after all, already in the nineteenth century, Gibbs' contributions to statistical mechanics as well as to vector analysis and Fourier series, and Hill's studies of Hill's equation, had put America on the applied mathematics map. The leading American analysts in the 1920s and 1930s were Birkhoff, renowned worldwide for his work in dynamics, and Wiener, a pioneer in the study of physical processes driven by chance influences such